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Mirrors > Home > ILE Home > Th. List > addnidpig | GIF version |
Description: There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) |
Ref | Expression |
---|---|
addnidpig | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ¬ (𝐴 +N 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7321 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | elni2 7326 | . . . 4 ⊢ (𝐵 ∈ N ↔ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) | |
3 | nnaordi 6522 | . . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝐵 → (𝐴 +o ∅) ∈ (𝐴 +o 𝐵))) | |
4 | nna0 6488 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴) | |
5 | 4 | eleq1d 2256 | . . . . . . . . 9 ⊢ (𝐴 ∈ ω → ((𝐴 +o ∅) ∈ (𝐴 +o 𝐵) ↔ 𝐴 ∈ (𝐴 +o 𝐵))) |
6 | nnord 4623 | . . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
7 | ordirr 4553 | . . . . . . . . . . . 12 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
8 | 6, 7 | syl 14 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴) |
9 | eleq2 2251 | . . . . . . . . . . . 12 ⊢ ((𝐴 +o 𝐵) = 𝐴 → (𝐴 ∈ (𝐴 +o 𝐵) ↔ 𝐴 ∈ 𝐴)) | |
10 | 9 | notbid 668 | . . . . . . . . . . 11 ⊢ ((𝐴 +o 𝐵) = 𝐴 → (¬ 𝐴 ∈ (𝐴 +o 𝐵) ↔ ¬ 𝐴 ∈ 𝐴)) |
11 | 8, 10 | syl5ibrcom 157 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ω → ((𝐴 +o 𝐵) = 𝐴 → ¬ 𝐴 ∈ (𝐴 +o 𝐵))) |
12 | 11 | con2d 625 | . . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 ∈ (𝐴 +o 𝐵) → ¬ (𝐴 +o 𝐵) = 𝐴)) |
13 | 5, 12 | sylbid 150 | . . . . . . . 8 ⊢ (𝐴 ∈ ω → ((𝐴 +o ∅) ∈ (𝐴 +o 𝐵) → ¬ (𝐴 +o 𝐵) = 𝐴)) |
14 | 13 | adantl 277 | . . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 +o ∅) ∈ (𝐴 +o 𝐵) → ¬ (𝐴 +o 𝐵) = 𝐴)) |
15 | 3, 14 | syld 45 | . . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝐵 → ¬ (𝐴 +o 𝐵) = 𝐴)) |
16 | 15 | expcom 116 | . . . . 5 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (∅ ∈ 𝐵 → ¬ (𝐴 +o 𝐵) = 𝐴))) |
17 | 16 | imp32 257 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) → ¬ (𝐴 +o 𝐵) = 𝐴) |
18 | 2, 17 | sylan2b 287 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → ¬ (𝐴 +o 𝐵) = 𝐴) |
19 | 1, 18 | sylan 283 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ¬ (𝐴 +o 𝐵) = 𝐴) |
20 | addpiord 7328 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵)) | |
21 | 20 | eqeq1d 2196 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 +N 𝐵) = 𝐴 ↔ (𝐴 +o 𝐵) = 𝐴)) |
22 | 19, 21 | mtbird 674 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ¬ (𝐴 +N 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 ∅c0 3434 Ord word 4374 ωcom 4601 (class class class)co 5888 +o coa 6427 Ncnpi 7284 +N cpli 7285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-irdg 6384 df-oadd 6434 df-ni 7316 df-pli 7317 |
This theorem is referenced by: (None) |
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